414 History of the Theory of Numbers. [Chap, xviii 
J. Perott' applied the theory of commutative groups to show that, 
if Q\,- ■ ■) Qn are primes, there exist at least n — 1 primes between g„ and 
M = qi- ■ Qn' 
T. J. Stieltjes^° expressed the product P of the primes 2, 3, . . ., p as a 
product AB of two factors in any way. Since A-{-B is not divisible by 
2, . . . , p, there exists a prime >p. 
J. Hacks" proved the existence of an infinitude of primes by use of his 
formula (Ch. XI, Hacks^^) for the number of integers ^m not divisible 
by a square. 
C. 0. Boije af Gennas^^ showed how to find a prime exceeding the nth 
primep„>2. Take P = 2'''3'''. . Pn*^, each ^^,^1. Express P as a product 
of relatively prime factors 5, P/8, where Q = P/8—8>l. Since Q is divisible 
by no prime ^ p„, it is a product of powers of primes qi'^pn+2. Take 5 so 
that Q< (p„+2)l Then Q is a prime. 
Axel Thue^^ proved that, if (l+n)*<2", there exist at least k-{-l 
primes <2''. 
J. Braun^'" noted that the sum of the inverses of the primes ^p is, for 
p^ 5, an irreducible fraction > 1 ; hence the numerator contains at least one 
prime >p. He attributed to Hacks a proof by means of 11(1 — l/p-)~^ = 
2s~^ = TT^/G ; the product would be rational if there were only a finite number 
of primes, whereas tt is irrational. 
E. Cahen^^ proved the ''identity of Euler" used by Kjonecker.^ 
Stormer-^^ gave a proof. 
A. Le\'y^^ took a product P of k of the first n primes Pi,. . ., p„ and 
the product Q of the remaining n — k. Then P+Q is either prime or has 
a prime factor >p„; like\\'ise for P — Q. If p„ is a prime such that p„+2 is 
composite, there exist at least n primes >p„, but ^l+PiP2- • Pn- When 
1 1 
± ± db 
Pi '" Pn 
is reduced to a simple fraction, the numerator has no factor in common with 
Pi . . .p„; hence there is a prime >Pn- He considered (pp. 242-4) the primes 
defiiied by a;(x — 1) — 1 for consecutive integers x. 
A. Auric^^ assumed that pi, . . . , pk give all the primes. Then the number 
of integers < 71 =npi°' is 
which is small in comparison with n, whence k increases indefinitely with n. 
»Amer. Jour. Math., 11, 1888, 9&-138; 13, 1891, 235-308, especially 303-5. 
"Annales fac. sc. de Toulouse, 4, 1890, 14, final paper. 
"Acta Math., 14, 1890-1, 335. 
'Hifversigt K. Sv. Vetenskaps-Akad. Forhand., Stockholm, .50, 1893, 469-471. 
"Archiv for Math, og Xatur., Kristiania, 19, 1897, No. 4, 1-5. 
'*^Das Fortschreitungsgesetz der Primzahlen durch eine transcendente Gleichung exakt 
dargestellt, WLss. Beilage Jahresbericht, Gymn., Trier, 1899, 96 pp. 
"filaments de la th^orie des nombres, 1900, 319-322. 
"Bull, de Math. £l6mentaires, 15, 1909-10, 33-34, 80-82. 
"L'interm^diaire des math., 22, 1915, 252. 
I 
